Finite-dimensionality in Tanaka theory
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 1, p. 75-90

In this paper we extend the Tanaka finiteness theorem and inequality for the number of symmetries to arbitrary distributions (differential systems) and provide several applications.

@article{AIHPC_2011__28_1_75_0,
     author = {Kruglikov, Boris},
     title = {Finite-dimensionality in Tanaka theory},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {28},
     number = {1},
     year = {2011},
     pages = {75-90},
     doi = {10.1016/j.anihpc.2010.10.001},
     zbl = {1260.58001},
     mrnumber = {2765511},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2011__28_1_75_0}
}
Kruglikov, Boris. Finite-dimensionality in Tanaka theory. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 1, pp. 75-90. doi : 10.1016/j.anihpc.2010.10.001. http://www.numdam.org/item/AIHPC_2011__28_1_75_0/

[1] I.M. Anderson, B. Kruglikov, Rank 2 distributions of Monge equations: symmetries, equivalences, extensions, arXiv:0910.5946 [math.DG] (2009) | MR 2824560

[2] E. Cartan, Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre, Ann. Sci. Éc. Norm. Super. (3) 27 (1910), 109-192 | JFM 41.0417.01 | MR 1509120

[3] E. Cartan, Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes, Bull. Soc. Math. France 42 (1914), 12-48 | JFM 45.0472.04

[4] A. Čap, K. Neusser, On automorphism groups of some types of generic distributions, Differential Geom. Appl. 27 (2009), 769-779 | MR 2552685 | Zbl 1190.58004

[5] B. Doubrov, I. Zelenko, On local geometry of nonholonomic rank 2 distributions, J. Lond. Math. Soc. 80 (2009), 545-566 | MR 2559116 | Zbl 1202.58002

[6] B. Doubrov, O. Radko, Graded nilpotent Lie algebras of infinite type, arXiv:1001.0379 [math.DG] (2010) | MR 2743103 | Zbl 1254.17025

[7] H. Goldschmidt, Integrability criteria for systems of nonlinear partial differential equations, J. Differential Geom. 1 no. 3 (1967), 269-307 | MR 226156 | Zbl 0159.14101

[8] A. Giaro, A. Kumpera, C. Ruiz, Sur la lecture correcte d'un résultat d'Élie Cartan, C. R. Acad. Sci. Paris Sér. A–B 287 no. 4 (1978), A241-A244 | MR 507769 | Zbl 0398.58003

[9] V. Guillemin, D. Quillen, S. Sternberg, The classification of the irreducible complex algebras of infinite type, J. Anal. Math. 18 (1967), 107-112 | MR 217130 | Zbl 0153.05903

[10] S. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag (1972) | MR 355886 | Zbl 0246.53031

[11] B. Kruglikov, V.V. Lychagin, Geometry of differential equations, D. Krupka, D. Saunders (ed.), Handbook on Global Analysis, Elsevier Sci. (2008), 725-771 | MR 2389645

[12] I.S. Krasilschik, V.V. Lychagin, A.M. Vinogradov, Geometry of Jet Spaces and Differential Equations, Gordon and Breach (1986) | Zbl 0722.35001

[13] A. Kumpera, D. Spencer, Lie Equations, vol. 1: General Theory, Princeton University Press, University of Tokyo Press (1972) | MR 380908 | Zbl 0258.58015

[14] M. Kuranishi, On E. Cartan's prolongation theorem of exterior differential systems, Amer. J. Math. 79 (1957), 1-47 | MR 81957 | Zbl 0077.29701

[15] T. Morimoto, Geometric structures on filtered manifolds, Hokkaido Math. J. 22 no. 3 (1993), 263-347 | MR 1245130 | Zbl 0801.53019

[16] T. Morimoto, N. Tanaka, The classification of the real primitive infinite Lie algebras, J. Math. Kyoto Univ. 10 no. 2 (1970), 207-243 | MR 268234 | Zbl 0211.05401

[17] R. Montgomery, M. Zhitomirskii, Geometric approach to Goursat flags, Ann. Inst. H. Poincaré Anal. Non Lineáire 18 no. 4 (2001), 459-493 | Numdam | MR 1841129 | Zbl 1013.58004

[18] A. Ottazzi, B. Warhurst, Contact and 1-quasiconformal maps on Carnot groups, arXiv:1001.3817 [math.DG] (2010) | MR 2917692 | Zbl 1254.22006

[19] J.-P. Serre, Lie algebras and Lie groups, Lectures given at Harvard University, 1964. | MR 218496

[20] I.M. Singer, S. Sternberg, On the infinite groups of Lie and Cartan, J. Anal. Math. 15 (1965), 1-114 | MR 217822 | Zbl 0277.58008

[21] D.C. Spencer, Overdetermined systems of linear partial differential equations, Bull. Amer. Math. Soc. (N.S.) 75 (1969), 179-239 | MR 242200 | Zbl 0185.33801

[22] S. Sternberg, Lectures on Differential Geometry, Prentice Hall, Englewood Cliffs (1964) | MR 193578 | Zbl 0129.13102

[23] N. Tanaka, On differential systems, graded Lie algebras and pseudo-groups, J. Math. Kyoto Univ. 10 no. 1 (1970), 1-82 | MR 266258 | Zbl 0206.50503

[24] B. Warhurst, Tanaka prolongation of free Lie algebras, Geom. Dedicata 130 (2007), 59-69 | MR 2365778 | Zbl 1283.17013

[25] E. Von Weber, Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen, Berichte Ges. Leipzig Math. Phys. Classe L (1898), 207-229 | JFM 29.0302.01

[26] K. Yamaguchi, Differential systems associated with simple graded Lie algebras, Adv. Stud. Pure Math. 22 (1993), 413-494 | MR 1274961 | Zbl 0812.17018

[27] I. Zelenko, On Tanaka's prolongation procedure for filtered structures of constant type, SIGMA 5 (2009), 094 | MR 2559667 | Zbl 1190.58007